Problems on market structure (chapters 10, 11, 12, and 13)
1. Bada Bing, Ltd., supplies standard 256 MB RAM chips to the U.S. computer and electronics industry.
Like the output of its competitors, Bada Bing’s chips must meet strict size, shape, and speed speci…cations. As a result, the chip-supply industry can be regarded as perfectly competitive. The total cost
and marginal cost functions for Bada Bing are:
T C = $1; 000; 000 + $20Q + $0:0001Q2
where Q is the number of chips produced.
a Calculate Bada Bing’s optimal output and pro…ts if chip prices are stable at $60 each.
Answer:
This industry is perfectly competitive so P = M R = $60. The …rm’s marginal cost is:
@T C
= M C = 20 + 0:0002Q
@Q
Setting M R = M C we have:
MR = MC
60 = 20 + 0:0002Q
40 = 0:0002Q
200; 000 = Q
Pro…t is T R
T C:
= TR TC
= (200; 000) 60
= $3; 000; 000
1; 000; 000 + 20 200; 000 + 0:0001 200; 0002
b Calculate Bada Bing’s optimal output and pro…ts if chip prices fall to $30 each.
Answer:
Again, because the industry is perfectly competitive we have P = M R = $30, and we still have
M C = 20 + 0:0002Q, so:
MR = MC
30 = 20 + 0:0002Q
10 = 0:0002Q
50; 000 = Q
And pro…t is now:
= (50; 000) 30
=
$750; 000
1; 000; 000 + 20 50; 000 + 0:0001 50; 0002
c If Bada Bing is typical of …rms in the industry, calculate the …rm’s long-run equilibrium output,
price, and economic pro…t levels.
Answer:
1
Recall that in long-run equilibrium the …rm will be producing a quantity equal to the minimum of the
…rm’s ATC curve because the …rm will be making zero pro…t. We have:
TC
=
AT C
=
$1; 000; 000 + $20Q + $0:0001Q2
1; 000; 000
+ 20 + 0:0001Q
Q
We also know that marginal cost intersects ATC at its minimum, so we set M C = AT C:
20 + 0:0002Q =
0:0001Q =
0:0001Q2
=
Q2
=
Q =
Q =
1; 000; 000
+ 20 + 0:0001Q
Q
1; 000; 000
Q
1; 000; 000
1; 000; 000
r0:0001
1; 000; 000
0:0001
100; 000
When Q = 100; 000, then we have:
P
P
P
P
P
= MC
= 20 + 0:0002Q
= 20 + 0:0002 100; 000
= 20 + 20
= 40
At a price of $40, the …rm should be making zero economic pro…t:
= (100; 000) 40
1; 000; 000 + 20 100; 000 + 0:0001 100; 0002
= 4; 000; 000 1; 000; 000 2; 000; 000 1; 000; 000
= 0
2. In class we discussed various reasons why a particular store may choose to remain open 24 hours.
Consider a rural Wal-Mart and an urban Wal-Mart.
Rural Wal-Mart
The rural Wal-Mart earns net revenues (net revenues here are total revenues minus product costs, so
the revenue earned on all product sales minus their cost) of $2,000 per day between the hours of 6am
and 11pm. It can also earn $100 in net revenues if it is open from 11pm to 6am. Electricity costs
$500 per day between 6am and 11pm if the store is open, and $100 per day if the store is closed during
those hours. Electricity costs $200 per day between the hours of 11pm and 6am if the store is open,
and $40 per day if the store is closed during those hours. If the store is open from 6am-11pm then it
must pay $1190 to cashiers and if the store is open from 11pm-6am it must pay another $84 to cashiers.
The table below gives the net revenues for the hours of operation as well as the electricity and cashiers
cost for these hours.
Open 6am-11pm Closed 6am-11pm Open 11pm-6am Closed 11pm-6am
Net Revenues $2000
$0
$100
$0
Electricity
$500
$100
$200
$40
Cashiers
$1190
$0
$84
$0
Urban Wal-Mart
The urban Wal-Mart earns net revenues (again, net revenues here are total revenues minus product
costs, so the revenue earned on all product sales minus their cost) of $5,000 per day between the hours
2
of 6am and 11pm. It can also earn $700 in net revenues if it is open from 11pm to 6am. Electricity
costs $1000 per day between 6am and 11pm if the store is open, and $200 per day if the store is closed
during those hours. Electricity costs $400 per day between the hours of 11pm and 6am if the store is
open, and $80 per day if the store is closed during those hours. If the store is open from 6am-11pm
then it must pay $2380 to cashiers and if the store is open from 11pm-6am it must pay another $168
to cashiers. The table below gives the net revenues for the hours of operation as well as the electricity
and cashiers cost for these hours.
Open 6am-11pm Closed 6am-11pm Open 11pm-6am Closed 11pm-6am
Net Revenues $5000
$0
$700
$0
Electricity
$1000
$200
$400
$80
Cashiers
$2380
$0
$168
$0
a What hours should the rural Wal-Mart be open? Explain.
Answer:
There are 4 cases that could happen:
a Open day, open night
b Open day, close night
c Close day, open night
d Close day, close night
The pro…t from (open day, open night) is: 2000
500
1190 + 100
The pro…t from (open day, close night) is: 2000
500
1190
40 = 270
The pro…t from (close day, open night) is:
100 + 100
200
84 =
The pro…t from (close day, close night) is:
100
40 =
200
84 = 126
284
140
Since the pro…t from opening during the day and closing at night is the greatest, the rural Wal-Mart
should be open from 6am–11pm. Note that the rural Wal-Mart cannot cover its TVC at night, so it
makes more money by shutting down.
b What hours should the urban Wal-Mart be open? Explain.
Answer:
There are 4 cases that could happen:
a Open day, open night
b Open day, close night
c Close day, open night
d Close day, close night
The pro…t from (open day, open night) is: 5000
1000
2380 + 700
The pro…t from (open day, close night) is: 5000
1000
2380
The pro…t from (close day, open night) is:
200 + 700
The pro…t from (close day, close night) is:
200
80 =
400
400
168 = 1752
80 = 1540
168 =
68
280
The urban Wal-Mart maximizes its pro…t by remaining open all 24 hours. Note that it can cover its
time period speci…c TVC with its time period speci…c revenue.
3
c The urban Wal-Mart has to pay people (stockers) to stock the shelves. These stockers can either
work from 6am-11pm (the day) or from 11pm-6am (the night). If they work during the day
then Wal-Mart must pay them $300. However, the stockers also cause some congestion in the
store during the day and the store loses $500 in net revenues during the day. If they work at
night Wal-Mart must pay them $500, but no net revenues are lost at night. One other factor to
consider is electricity. Whenever the stockers work Wal-Mart must pay full electricity regardless
of whether or not the store is actually open (so it must pay $400 in electricity if the stockers work
at night even if the store is closed). Assuming this urban Wal-Mart must hire stockers, answer
the following questions:
i Will the urban Wal-Mart be making a pro…t or loss during the night hours if it hires the stockers
to work at night and it opens?
Answer:
The urban Wal-Mart will be making a loss if it hires the stockers to work at night and it opens. It
will have to pay $500 for the stockers, $400 for the electricity, and $168 for the cashiers. The store
does have net revenues of $700 though, so its loss is $368.
ii If the urban Wal-Mart hires the stockers to work at night should they open? Explain.
Answer:
The urban Wal-Mart should open if it hires the stockers to work at night. We already know (from
part a) that the store will lose $368 if it hires the stockers to work at night and it opens. If it hires
the stockers to work at night and it closes, it will lose $900, $500 from paying the stockers and $400
from keeping the lights on.
iii If Wal-Mart is a business intent on maximizing its pro…ts, during which hours should it hire the
stockers, and what should the store’s hours of business be? Explain, making sure to reference
the pro…t level of your decision as well as the pro…t level of other decisions that could have
been made.
Answer:
The urban store should open all 24 hours and it should hire its stockers to work at night. The store
will make a pro…t of 5000 1000 2380 + 700 400 168 500 = 1252 if it follows this business plan.
There are a number of alternative plans the store could use, but we know one thing: if the store hires
the stockers to work nights, they will open. We know this because parts a and b of the question show
us that opening when the stockers work at night loses less money than being closed. So, here are the
8 cases, with their pro…t levels:
1 Open day (stockers), open night: 5000 1000 2380 300 500 + 700 400 168 = 952
2 Open day, open night (stockers): 5000 1000 2380 + 700 400 168 500 = 1252
3 Open day (stockers), close night: 5000 1000 2380 300 500 80 = 740
4 Open day, close night (stockers): 5000 1000 2380 400 500 = 720
5 Close day (stockers), open night: 1000 300 + 700 400 168 = 1168
6 Close day, open night (stockers): 200 + 700 400 168 500 = 568
7 Close day (stockers), close night: 1000 300 80 = 1380
8 Close day, close night (stockers): 200 400 500 = 1100
Notice that the highest pro…t is 1252 when the store is open 24 hours and hires the stockers to work
at night. Even though the store makes a loss by hiring the stockers to work at night and opening
(this is the $368 loss in part b), this loss is less than the loss that would occur if it closed, and it is
less than the loss that would occur if the store hired the stockers to work during the day (they would
save $200 in stocker costs, but would lose another $500 in revenues from congestion, which is why the
pro…t from opening 24 hours and hiring the stockers during the day is only $952). So Wal-Mart may
4
actually be earning a loss on its night-time operation, but that may be the best time for the store to
have its stockers work.
3. Bates Gill is the sole developer of underwater operating systems. His …rm is protected by considerable
barriers to entry. Gill faces the following inverse demand function for his underwater operating systems:
P (Q) = 4320
12Q
His total cost function is:
T C = 4Q2 + 200; 000
a Write down Gill’s M R function solely as a function of quantity.
Answer:
We know that:
MR
=
TR
MR
=
=
@T R
@Q
(4320 12Q) Q
4320 24Q
b Find Gill’s pro…t-maximizing quantity.
Answer:
All we need to do is set M R = M C.
4320
24Q = 8Q
4320 = 32Q
135 = Q
So Gill’s pro…t-maximizing quantity is 135.
c Find Gill’s price at the pro…t-maximizing quantity.
Answer:
To …nd the price at the pro…t-maximizing quantity simply plug the pro…t-maximizing quantity into
the inverse demand function:
P (135) = 4320
12 135 = 2700
So the price is $2700.
d What are Gill’s pro…ts at the pro…t-maximizing price and quantity?
Answer:
To …nd the pro…t simply subtract total cost from total revenue.
= TR
= 2700 135
TC
2
4 (135) + 200000 = 91600
5
So Gill’s pro…t at the pro…t-maximizing level is $91; 600.
Adding the government
The government realizes that Gill is a monopolist and that “considerable”deadweight loss is being created in the underwater operating systems market. Use the functions above to answer these questions.
e Draw a picture (just a rough sketch, not necessarily to scale) that illustrates Gill’s pro…t-maximizing
price and quantity as well as the quantity and the price that would be charged if the market were
to operate e¢ ciently (with no deadweight loss).
Answer:
f Find the price and quantity that correspond that will make this market completely e¢ cient (no
deadweight loss).
Answer:
Again, the reason I asked the question about the graph above was to aid you in answering this question.
The socially e¢ cient quantity occurs at the intersection of the demand curve and the M C. So, setting
P (Q) = M C we can …nd the socially e¢ cient quantity.
4320
12Q = 8Q
4320 = 20Q
216 = Q
Now that we have the socially e¢ cient quantity, we can …nd the socially e¢ cient price by plugging
that quantity into the inverse demand function:
P (216) = 4320
12 216 = 1728
So the socially e¢ cient price in this market is $1; 728.
6
g Suppose that Gill had not yet entered into this market, but was merely planning to enter into the
market. If the government were to tell Gill ahead of time that they would regulate his price at
the e¢ cient level, would Gill enter this market? Explain.
Answer:
No, Gill would not enter this market. There are a variety of methods to answer this question. One
is to look at Gill’s pro…t at the socially e¢ cient price and quantity level:
= TR
= 1728 216
TC
2
4 (216) + 200000 =
13376
Since Gill’s pro…t is less than 0, he would not enter this market.
Another method would be to compare the price and ATC at the socially e¢ cient quantity. We know
the price is 1728, and the ATC is given by:
AT C (Q) = 4Q +
AT C (216) = 4 (216) +
200; 000
Q
200; 000
= 1; 789:93
216
Since P < AT C at this quantity, Gill will not enter.
4. The market for a rare strain of apples is a perfectly competitive market, with 150 identical sellers. The
market is a constant cost industry. Each seller has the following costs:
TC
=
4q 2
AT C
=
4q
MC
=
8q
4q + 144
144
4+
q
4
The supply and demand conditions in the market are such that the market price is $52 per apple (I
told you they were rare).
a Find the pro…t-maximizing quantity for an individual …rm in this market.
Answer:
The easiest way to …nd the pro…t-maximizing quantity was to …nd where M R = M C. Since this is a
perfectly competitive market, M R = P , which means that M R = 52. Setting this equal to M C gives:
52 = 8q
4
56 = 8q
7=q
So each …rm should produce 7 units in this market.
You could also have tried the table method that was used on the homework. It might not have been
too bad for this question since the pro…t-maximizing quantity was 7 units, but if you tried this method
on Bates Gill number 2 it probably did not work so well.
7
b What is the pro…t-level at the pro…t-maximizing quantity for an individual …rm in this market.
Answer:
To …nd the pro…t-level for this …rm, simply …nd the total revenue at the pro…t-maximizing quantity
and the total cost at the pro…t-maximizing quantity, and subtract total cost from total revenue.
TR = P
Q = 52 7 = 364
2
T C = 4 (7)
TR
4 (7) + 144 = 312
T C = 364
312 = 52
So the pro…t at the pro…t-maximizing quantity is $52.
c What is the total MARKET quantity in this industry?
Answer:
To …nd the total market quantity, simply …nd the product of the quantity produced by each …rm and
the total number of …rms. This method works because the …rms are identical. So 150 7 = 1050.
d Using the graphs below, draw a picture of the competitive market and …rm in LR equilibrium. Be
sure to label your graphs, and to include the …rm’s AT C, M C, and M R.
Answer:
e Find the price that would correspond to the apple market being in a LR equilibrium. Remember,
the apple market is a constant cost industry.
Answer:
The reason that I asked the question about the graph (part d) was to provide you with an aid to
answering this question. Looking at the graph in part d, notice that the LR equilibrium occurs at the
quantity where P = M C = AT C. Because the price is the variable we wish to solve for we cannot
8
use it to …nd the quantity. However, we can use the fact that the quantity the …rm produces in LR
equilibrium is the one that corresponds to the point where M C = AT C. Setting M C = AT C gives:
8q
4 = 4q
4q =
4+
144
q
144
q
4q 2 = 144
q 2 = 36
q=
6
Now, we can’t really have a quantity of ( 6), so the answer has to be q = 6. Now we can …nd the
price in the market by setting P = M C (or P = AT C) when q = 6.
P = MC
P = 8 (6)
4 = 44
OR:
P = AT C
P = 4 (6)
4+
144
= 44
6
Either way, the price in the market is $44. If you really wanted to double check your work, you can
…nd the pro…t-level at this price and quantity pair. T R = P Q = 44 6 = 264. T C = 4q 2 4q +144 =
2
4 (6)
4 (6) + 144 = 264. Because pro…t equals zero, this …rm is in LR equilibrium.
f Explain why the fact that the apple market is a constant cost industry is a useful assumption in
solving part e. If the apple market were a decreasing cost industry, how would this a¤ect the
resulting LR equilibrium price (would it be higher or lower than the price you found in part e
and why).
Answer:
In the initial structure of the problem, with P = $52, the …rms in the market were making economic
pro…ts. This means that more …rms will enter into the market, which will decrease the price and
increase the total market quantity. If the constant cost industry assumption were not made, then
the …rms’cost curves would shift as the total market quantity increases (the cost curves would shift
downward in a decreasing cost industry and upward in an increasing cost industry). And if the …rms’
cost curves are shifting, there is no possible way to answer number 5 because you don’t know how
much they would shift with the increase in total market quantity.
If this were a decreasing cost industry then the price would be lower than the one in question number
5, as the …rms’cost curves would shift downward as market quantity increases.
9
5. To many upscale homeowners, no other ‡ooring o¤ers the warmth, beauty, and value of wood. New
technology in stains and …nishes call for regular cleaning that takes little more than sweeping and/or
vacuuming, with occasional use of a professional wood ‡oor cleaning product. Wood ‡oors are also
ecologically friendly because wood is both renewable and recyclable. Buyers looking for traditional
oak, rustic pine, trendy mahogany, or bamboo can choose from a wide assortment.
At the wholesale level, wood ‡ooring is a commodity-like product sold with rigid product speci…cations.
Price competition is ferocious among hundreds of domestic manufacturers and importers. Assume that
market supply and demand conditions for mahogany wood ‡ooring are:
QS
QD
=
=
10 + 2P
320 4P
where Q is output in square yards of ‡oor covering (000), and P is the market price per square yard.
a Calculate the equilibrium price/output solution before and after imposition of a $9 per unit tax on
suppliers.
Answer:
Calculating the before-tax equilibrium price and quantity we simply need to set QS = QD :
10 + 2P
6P
P
=
=
=
320
330
55
4P
When P = 55, then Q = 100 (actually 100; 000 given the units of measurement).
When the tax is imposed, we alter the supply function by subtracting the amount of the tax from the
price:
QS
QS
QS
=
=
=
10 + 2 (P
10 + 2 (P
28 + 2P
)
9)
We subtract the amount of the tax because the price the suppliers receive is P
QS = QD again:
28 + 2P
6P
P
= 320
= 348
= 58
. Now we set
4P
When P = 58, Q = 88 (actually 88; 000).
b Calculate the deadweight loss to taxation caused by imposition of the $9 per unit tax. How much
of this deadweight loss was su¤ered by consumers versus producers? Explain.
Answer:
There is a $9 tax and a loss of 12,000 units sold in the market due to the tax. Because the supply and
demand functions are linear, the deadweight loss will be given by a triangle with a "base" equal to 9
and a "height" equal to 12,000. Using 21 base height we have that the total deadweight loss is:
DW L =
DW L =
1
9 12; 000
2
54; 000
The easiest way to determine the burden on the consumers and the burden on the suppliers is to look
at the after-tax price in relation to the before-tax price. The after-tax price is $3 higher than the
before-tax price, so consumers are paying 13 of the $9 tax. Thus the consumers su¤er a DWL of 18,000,
while the producers (who are bearing $6 of the $9 tax) are su¤ering a DWL of 36,000.
10
6. Each year, about 9 billion bushels of corn are harvested in the United States. The average market
price of corn is a little over $2 per bushel, but costs farmers about $3 per bushel. Tax payers make
up the di¤erence. Under the 2002 $190 billion, 10-year farm bill, American taxpayers will pay farmers
$4 billion a year to grow even more corn, despite the fact that every year the United States is faced
with a corn surplus. Growing surplus corn also has unmeasured environmental costs. The production
of corn requires more nitrogen fertilizer and pesticides than any other agricultural crop. Runo¤ from
these chemicals seeps down into the groundwater supply, and into rivers and streams. Ag chemicals
have been blamed for a 12,000-square-mile dead zone in the Gulf of Mexico. Overproduction of corn
also increases U.S. reliance on foreign oil.
To illustrate some of the cost in social welfare from agricultural price supports, assume the following
market supply and demand conditions for corn:
QS
QD
=
=
5; 000 + 5; 000P
10; 000 2; 500P
where Q is output in bushels of corn (in millions), and P is the market price per bushel.
a Calculate the equilibrium price/output solution.
Setting QS = QD :
5; 000 + 5; 000P
7; 500P
P
= 10; 000
= 15; 000
= 2
2; 500P
When P = 2, then Q = 5; 000 (in millions).
b Calculate the amount of surplus production the government will be forced to buy if it imposes a
price ‡oor of $2.50 per bushel.
Answer:
If P = 2:50, then we have:
QS
QD
=
=
5; 000 + 5; 000 2:5 = 7; 500 (in millions)
10; 000 2; 500 2:5 = 3; 750 (in millions)
So there will be a surplus of 3; 750 (in millions)
7. Calvin’s Barber Shops, Inc. has a monopoly on barbershop services provided on the south side of
Chicago because of restrictive licensing requirements, and not because of superior operating e¢ ciency.
As a monopoly, Calvin’s provides all industry output. Assume T C = 20Q.
Assume that:
P = $80
$0:0008Q
Where P is price per unit and Q is total …rm output (haircuts).
a Calculate the monopoly pro…t maximizing price/output combination, as well as pro…ts for the monopolist.
Answer:
To …nd the pro…t-maximizing price and output combination, set M R = M C:
TR
MR
= ($80 $0:0008Q) Q
= 80 0:0016Q
11
Now:
80
When Q = 37; 500, we have P = 50.
(50 20) = 1; 125; 000.
MR
0:0016Q
60
Q
= MC
= 20
= 0:0016Q
= 37; 500
Monopoly pro…ts are given by Q
(P
AT C) = 37; 500
b What is the competitive market long-run equilibrium (price and quantity of haircuts)?
Answer:
For the perfectly competitive outcome, we need P = M C = AT C. Both M C and AT C equal 20, so:
$80
$0:0008Q = 20
60 = 0:0008Q
Q = 75; 000
c Discuss the “monopoly problem” from a social perspective in this instance.
Answer:
The problem of monopoly is that prices are higher and output is lower than the competitive equilibrium.
In this instance, output is half of what it would be in a competitive market, and prices are 2.5 times
higher.
8. Consider a simultaneous quantity choice (Cournot) game between 2 …rms. Each …rm chooses a
quantity, q1 and q2 respectively. The inverse market demand function is given by P (Q) = 1434 2 Q,
2
where Q = q1 + q2 . Firm 1 has total cost function T C (q1 ) = 3 (q1 ) and Firm 2 has total cost
2
function T C (q2 ) = 12 (q2 )
12 q2 . Each …rm wishes to maximize pro…t.
a Set up the pro…t function for Firm 1 and Firm 2. Remember, this is a quantity choice game.
Answer:
Pro…t functions are simply total revenue for the …rm minus total cost. For each …rm their respective
total revenue is given by the product of their quantity and the market price. For each …rm their
respective total cost is given by their total cost function. So we have:
1
2
=
=
(1434
(1434
2q1
2q1
2q2 ) q1
2q2 ) q2
3q12
12q22
12q2
b Find the best response functions for Firms 1 and 2.
Answer:
Firm 1 maximizes:
1
=
(1434
2q1
2q2 ) q1
@ 1
= 1434 4q1 2q2 6q1
@q1
0 = 1434 10q1 2q2
10q1 = 1434 2q2
1434 2q2
q1 =
10
717 q2
q1 =
5
12
3q12
Technically, Firm 1’s best response function is q1 = M ax 0; 7175
q2
.
Firm 2 maximizes:
=
2
(1434
2q1
2q2 ) q2
@ 2
= 1434 2q1
@q2
0 = 1446 2q1
28q2 = 1446 2q1
1446 2q1
q2 =
28
723 q1
q2 =
14
4q2
12q22
12q2
24q2 + 12
28q2
Technically, Firm 2’s best response function is q2 = M ax 0; 72314 q1 .
c Find the equilibrium to this game.
Answer:
To …nd the equilibrium to the game simply substitute one best response function into the other:
q1
q1
5q1
70q1
69q1
q1
=
=
717
q2
5
717
723 q1
14
5
723 q1
= 717
14
= 10038 723 + q1
= 9315
= 135
Now to …nd Firm 2’s quantity simply use q1 = 135 in Firm 2’s best response function:
q2
=
q2
=
q2
=
723 q1
14
723 135
14
42
So the equilibrium to this game is q1 = 135 and q2 = 42.
d Find the (1) total market quantity, (2) price, and (3) pro…t for each …rm.
Answer:
The market quantity is Q = q1 + q2 , so Q = 177. The market price is P (Q) = 1434
Each …rm’s pro…t is:
1
1
1
2
2
2
= P q1 T C (q1 )
= 1080 135 3 1352
= 91125
and
= P q1 T C (q2 )
= 1080 42
12 422
= 24696
13
12 42
2Q, so P = 1080.
e Assume Firm 1 is the only producer in the market now. Find Firm 1’s monopoly (1) quantity, (2)
price, and (3) pro…t.
Answer:
If Firm 1 is the only producer then Firm 1 is a monopolist so that q2 = 0. There are many ways to
go about this, the most direct being to set up Firm 1’s pro…t function (assuming q2 = 0) and solve for
the monopoly quantity:
1
=
(1434
2q1 ) q1
3q12
d 1
= 1434 4q1 6q1
dq1
0 = 1434 10q1
10q1 = 1434
q1 = 143:4
We now know that Firm 1’s monopoly quantity is 143:4 (we could also have used Firm 1’s best response
function from part b to …gure this out), with price being equal to
P (Q) = 1434 2Q
P (143:4) = 1434 2 143:4
P = 1147:2
Now knowing both P and q1 we can …nd Firm 1’s pro…t:
1
= P
q1
T C (q1 )
2
1
1
1
= 1147:2 143:4 3 (143:4)
= 164508:48 61690:68
= 102817:8
9. Consider two …rms who compete by simultaneously choosing prices (a Bertrand game). If …rms 1 and
2 choose prices p1 and p2 , respectively, the quantity that consumers demand from …rm i is
qi (pi ; pj ) = a
pi + bpj , with 0 < b < 2.
Assume that there are no …xed costs and that marginal cost is constant and equal to c, where a > c > 0.
Prices must be nonnegative (p1 0, p2 0) and …rms wish to maximize pro…t.
a Find the best response functions for …rms 1 and 2.
Answer:
This problem is a straightforward maximization problem. Using the case-by-case analysis approach
won’t work here. Firm 1’s maximization problem is:
max
p1 0
max
p1 0
The …rst-order condition is:
1
= q1 (p1 ; p2 ) p1
1
or
= (a
c q1 (p1 ; p2 )
p1 + bp2 ) p1
@ 1
=a
@p1
c (a
2p1 + bp2 + c = 0
Firm 1’s best response function is then:
p1 =
a + bp2 + c
2
14
p1 + bp2 )
Firm 2’s problem is similar:
max
p2 0
max
p2 0
2
= q2 (p1 ; p2 ) p2
2
or
= (a
c q2 (p1 ; p2 )
p2 + bp1 ) p2
c (a
p2 + bp1 )
and yields a similar …rst-order condition:
@ 2
=a
@p2
2p2 + bp1 + c = 0
Firm 2’s best response function is then:
p2 =
a + bp1 + c
2
b Find the equilibrium to this game.
Answer:
The equilibrium to this game is a price, p1 , for Firm 1 and a price, p2 , for Firm 2. There are 2
equations (the best response functions) and 2 unknowns (p1 and p2 ), so we need to solve for p1 and p2 .
p1
=
p2
=
a + bp2 + c
2
a + bp1 + c
2
Substituting leads to:
2p1
4p1
4p1
b2 p1
p1
p1
p1
p1
= a+b
a + bp1 + c
2
+c
2a + ab + b2 p1 + bc + 2c
2a + ab + bc + 2c
2a + ab + bc + 2c
=
4 b2
(2a + ab) + (bc + 2c)
=
(2 b) (2 + b)
a (2 + b) + c (2 + b)
=
(2 b) (2 + b)
a+c
=
2 b
=
=
Now substituting back into Firm 2’s best response function, we get:
a+b
(2
(4
(4
p2
=
2p2
=
b) 2p2
2b) p2
2b) p2
=
=
=
p2
=
p2
=
a+c
2 b
+c
2
a c
a+b
+c
2 b
(2 b) a + ba bc + c (2
2a ba + ba bc + 2c bc
2a + 2c
2 (a + c)
2 (2 b)
a+c
2 b
15
b)
So the equilibrium is:
p1
=
p2
=
a+c
2 b
a+c
2 b
c Explain why b < 2.
Answer:
Looking at the Nash equilibrium strategies, we see that if b > 2 then the best response by a …rm would
be a negative price, which is not allowable.
10. Consider a simultaneous quantity choice game between 2 …rms. Each …rm chooses a quantity, q1
and q2 respectively. The inverse market demand function is given by P (Q) = 1980 6Q, where
Q = q1 + q2 . Firm 1 has total cost function T C (q1 ) = 24q1 + 624 and Firm 2 has total cost function
2
T C (q2 ) = 18 (q2 ) + 12q2 + 16. The pro…t functions for each …rm are:
1
=
(1980
6q1
6q2 ) q1
2
=
(1980
6q1
6q2 ) q2
(24q1 + 624)
2
18 (q2 ) + 12q2 + 16
a Find the best response functions for Firms 1 and 2.
Answer:
For Firm 1 we have:
1
1
6q12
6q12
= 1980q1
= 1956q1
@ 1
= 1956 12q1
@q1
0 = 1956 12q1
12q1 = 1956 6q2
1956 6q2
q1 =
12
6q1 q2
6q1 q2
24q1
624
624
6q2
6q2
For Firm 2 we have:
2
2
= 1980q2
= 1968q2
6q1 q2 6q22 18q22
24q22 6q1 q2 16
@ 2
= 1968 48q2
@q2
0 = 1968 48q2
48q2 = 1968 6q1
1968 6q1
q2 =
48
12q2
16
6q1
6q1
So the best response functions for Firm 1 and 2 are, respectively: q1 = 195612 6q2 and q2 = 196848 6q1 .
Technically, the best responses are: q1 = M ax 0; 195612 6q2 and q2 = M ax 0; q2 = 196848 6q1 because
neither …rm can produce a negative quantity.
b Find the Nash equilibrium to this game.
Answer:
16
Using the best response functions we have:
q1
=
q1
=
12q1
=
12q1
=
96q1
90q1
q1
=
=
=
1956 6q2
12
1956 6 196848 6q1
12
1968 6q1
1956 6
48
1968 6q1
1956
8
15648 1968 + 6q1
13680
152
Now using that q1 = 152 we have:
1968 6q1
48
1968 6 152
=
48
= 22
q2
=
q2
q2
So the equilibrium to this game is: q1 = 152 and q2 = 22.
c Find (1) the total market quantity, (2) the market price, and (3) each …rm’s pro…t.
Answer:
The total market quantity is Q = q1 + q2 = 174.
The price is P (Q) = 1980
6Q = 1980
6 174 = 936.
Firm 1’s pro…t is:
1
1
1
1
= P q1 (24q1 + 624)
= 936 152 (24 152 + 624)
= 142272 3648 624
= 138000
Firm 2’s pro…t is:
2
2
2
2
2
= P
q2
18q22 + 12q2 + 16
= 936 22
18 222 + 12 22 + 16
= 20592 (8712 + 264 + 16)
= 20592 8992
= 11600
17